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| In [[mathematics]], the '''Lindelöf hypothesis''' is a conjecture by Finnish mathematician [[Ernst Leonard Lindelöf]] (see {{harvtxt|Lindelöf|1908}}) about the rate of growth of the [[Riemann zeta function]] on the critical line that is implied by the [[Riemann hypothesis]].
| | == a move Godhead directly the speed of light == |
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| It says that, for any ''ε'' > 0,
| | 'Block,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_72.htm ゴルフ用サングラスオークリー]!'<br><br>Yan Emperor whispered.<br><br>immediately to the main hall where he was core to spread out in all directions, a full range of spatial fluctuations of hundreds of light years away completely locked.<br><br>'Although I Yan Protoss space coordinates corresponding long concealed to prevent in case, I add some means,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_36.htm オークリー サングラス 調光], even if you invite chaos lord to crack,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_45.htm オークリー サングラス 人気ランキング], it is difficult to crack open short side.' Yan Emperor smiled,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_65.htm オークリー サングラス], a move Godhead directly the speed of light, the shuttle began to cast into the dark universe means it.<br><br>,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_18.htm オークリー スポーツサングラス]......<br><br>Whoosh! Whoosh! Whoosh!<br><br>Feng Luo a decision Teleport times, approaching Yan Protoss lair.<br><br>'coming to!' Luo Feng increasingly expect.<br><br>'ah?' Luo Feng suddenly stopped, looked at the front frown void,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_2.htm ロードバイク サングラス オークリー], the void has some Starshards floating, 'spatial fluctuations even completely blocked? how is it?'<br><br>'polytheism ancestral lair,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_64.htm オークリー 激安 サングラス], the lake is unable to teleport colorful aurora,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_18.htm オークリー スポーツサングラス], only some of the island's area Bauhinia island can blink |
| | | 相关的主题文章： |
| :<math>\zeta\left(\frac12 + it\right) \mbox{ is }\mathcal{O}(t^\varepsilon),</math> | | <ul> |
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| as ''t'' tends to infinity (see [[O notation]]). Since ''ε'' can be replaced by a smaller value, we can also write the conjecture as, for any positive ''ε'',
| | <li>[http://wangdajie.com.cn/plus/feedback.php?aid=157 http://wangdajie.com.cn/plus/feedback.php?aid=157]</li> |
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| :<math>\zeta\left(\frac12 + it\right) \mbox{ is }o(t^\varepsilon).</math>
| | <li>[http://www.xinyuzuowen.com/plus/feedback.php?aid=219 http://www.xinyuzuowen.com/plus/feedback.php?aid=219]</li> |
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| ==The μ function==
| | <li>[http://www.dzkunlun.com/plus/feedback.php?aid=380 http://www.dzkunlun.com/plus/feedback.php?aid=380]</li> |
| If σ is real, then μ(σ) is defined to be the [[infimum]] of all real numbers ''a'' such that ''ζ''(''σ'' + ''iT'') = O(''T''<sup> ''a''</sup>). It is trivial to check that ''μ''(''σ'') = 0 for ''σ'' > 1, and the [[functional equation]] of the zeta function implies that μ(''σ'') = ''μ''(1 − ''σ'') − ''σ'' + 1/2. The [[Phragmen–Lindelöf theorem]] implies that μ is a [[convex function]]. The Lindelöf hypothesis states μ(1/2) = 0, which together with the above properties of ''μ'' implies that ''μ''(''σ'') is 0 for ''σ'' ≥ 1/2 and 1/2 − σ for ''σ'' ≤ 1/2.
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| | | </ul> |
| Lindelöf's convexity result together with ''μ''(1) = 0 and ''μ''(0) = 1/2 implies that 0 ≤ ''μ''(1/2) ≤ 1/4. The upper bound of 1/4 was lowered by [[G. H. Hardy|Hardy]] and [[J. E. Littlewood|Littlewood]] to 1/6 by applying [[Hermann Weyl|Weyl]]'s method of estimating [[exponential sum]]s to the [[approximate functional equation]]. It has since been lowered to slightly less than 1/6 by several authors using long and technical proofs, as in the following table:
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| {| class="wikitable"
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| |-
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| ! μ(1/2) ≤
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| ! μ(1/2) ≤
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| ! Author
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| |-
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| | 1/4
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| | 0.25
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| | {{harvtxt|Lindelöf|1908}}
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| |Convexity bound
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| |-
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| | 1/6
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| | 0.1667
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| | {{harvtxt|Hardy|Littlewood|?}}
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| |-
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| |163/988
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| |0.1650
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| |{{harvtxt|Walfisz|1924}}
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| |-
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| |27/164
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| |0.1647
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| |{{harvtxt|Titchmarsh|1932}}
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| |-
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| |229/1392
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| |0.164512
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| |{{harvtxt|Phillips|1933}}
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| |-
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| |0.164511
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| |{{harvtxt|Rankin|1955}}
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| |-
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| |19/116
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| |0.1638
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| |{{harvtxt|Titchmarsh|1942}}
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| |-
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| |15/92
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| |0.1631
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| |{{harvtxt|Min|1949}}
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| |-
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| |6/37
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| |0.16217
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| |{{harvtxt|Haneke|1962}}
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| |-
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| |173/1067
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| |0.16214
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| |{{harvtxt|Kolesnik|1973}}
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| |-
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| |35/216
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| |0.16204
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| |{{harvtxt|Kolesnik|1982}}
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| |-
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| |139/858
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| |0.16201
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| |{{harvtxt|Kolesnik|1985}}
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| |-
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| |32/205
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| |0.1561
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| |{{harvs|txt|last=Huxley|year1=2002|year2=2005}}
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| |}
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| ==Relation to the Riemann hypothesis==
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| {{harvtxt|Backlund|1918–1919}} showed that the Lindelöf hypothesis is equivalent to the following statement about the zeros of the zeta function: for every ''ε'' > 0, the number of zeros with real part at least 1/2 + ''ε'' and imaginary part between ''T'' and ''T'' + 1 is o(log(''T'')) as ''T'' tends to infinity. The Riemann hypothesis implies that there are no zeros at all in this region and so implies the Lindelöf hypothesis. The number of zeros with imaginary part between ''T'' and ''T'' + 1 is known to be O(log(''T'')), so the Lindelöf hypothesis seems only slightly stronger than what has already been proved, but in spite of this it has resisted all attempts to prove it and is very hard.
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| ==Means of powers of the zeta function==
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| The Lindelöf hypothesis is equivalent to the statement that
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| :<math>\int_0^T|\zeta(1/2+it)|^{2k}\,dt = O(T^{1+\varepsilon})</math>
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| for all positive integers ''k'' and all positive real numbers ε. This has been proved for ''k'' = 1 or 2, but the case ''k'' = 3 seems much harder and is still an open problem.
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| There is a much more precise conjecture about the asymptotic behavior of this integral: it is believed that
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| :<math>\int_0^T|\zeta(1/2+it)|^{2k} \, dt = T\sum_{j=0}^{k^2}c_{k,j}\log(T)^{k^2-j} + o(T)</math>
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| for some constants ''c''<sub>''k'',''j''</sub>. This has been proved by Littlewood for ''k'' = 1 and by {{harvtxt|Heath-Brown|1979}} for ''k'' = 2
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| (extending a result of {{harvtxt|Ingham|1926}} who found the leading term).
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| {{harvtxt|Conrey|Ghosh|1998}} suggested the value <math>(42/9!)\prod_ p \left((1-p^{-1})^4(1+4p^{-1}+p^{-2})\right)</math> for the leading coefficient when ''k'' is 6, and {{harvtxt|Keating|Snaith|2000}} used [[random matrix theory]] to suggest some conjectures for the values of the coefficients for higher ''k''. The leading coefficients are conjectured to be the product of an elementary factor, a certain product over primes, and the number of n by n [[Young tableaux]] given by the following sequence:
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| * 1, 1, 2, 42, 24024, 701149020, {{OEIS|id=A039622}}.
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| ==Other consequences==
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| Denoting by ''p''<sub>''n''</sub> the ''n''-th prime number, a result by [[Albert Ingham]], shows that the Lindelöf hypothesis implies that, for any ''ε'' > 0,
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| :<math>p_{n+1}-p_n\ll p_n^{1/2+\varepsilon}\,</math>
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| if ''n'' is [[sufficiently large]]. However, this result is much worse than that of the large [[prime gap]] conjecture.
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| ==References==
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| *{{citation|last=Backlund|first= R. |title=Über die Beziehung zwischen Anwachsen und Nullstellen der Zeta-Funktion|journal= Ofversigt Finska Vetensk. Soc. |volume=61|issue=9|year=1918–1919}}
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| *{{Citation | last1=Conrey | first1=J. B. | last2=Farmer | first2=D. W. | last3=Keating | first3=Jonathan P. | last4=Rubinstein | first4=M. O. | last5=Snaith | first5=N. C. | title=Integral moments of L-functions | doi=10.1112/S0024611504015175 | mr=2149530 | year=2005 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=91 | issue=1 | pages=33–104}}
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| * {{Citation | last1=Conrey | first1=J. B. | last2=Farmer | first2=D. W. | last3=Keating | first3=Jonathan P. | last4=Rubinstein | first4=M. O. | last5=Snaith | first5=N. C. | title=Lower order terms in the full moment conjecture for the Riemann zeta function | doi=10.1016/j.jnt.2007.05.013 | mr=2419176 | year=2008 | journal=[[Journal of Number Theory]] | issn=0022-314X | volume=128 | issue=6 | pages=1516–1554}}
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| *{{Citation | last1=Conrey | first1=J. B. | last2=Ghosh | first2=A. | title=A conjecture for the sixth power moment of the Riemann zeta-function | doi=10.1155/S1073792898000476 | mr=1639551 | year=1998 | journal=International Mathematics Research Notices | issn=1073-7928 | issue=15 | pages=775–780 | volume=1998}}
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| *{{Citation | last1=Edwards | first1=H. M. | authorlink = Harold Edwards (mathematician) | title=Riemann's Zeta Function | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-41740-0 | mr=0466039 | year=1974}}
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| *{{Citation | authorlink=Roger Heath-Brown | last1=Heath-Brown | first1=D. R. | title=The fourth power moment of the Riemann zeta function | doi=10.1112/plms/s3-38.3.385 | mr=532980 | year=1979 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=38 | issue=3 | pages=385–422}}
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| *{{Citation | last1=Huxley | first1=M. N. | title=Number theory for the millennium, II (Urbana, IL, 2000) | publisher=[[A K Peters]] | mr=1956254 | year=2002 | chapter=Integer points, exponential sums and the Riemann zeta function | pages=275–290}}
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| *{{Citation | last1=Huxley | first1=M. N. | title=Exponential sums and the Riemann zeta function. V | doi=10.1112/S0024611504014959 | mr=2107036 | year=2005 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=90 | issue=1 | pages=1–41}}
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| *{{citation|first=A. E. |last=Ingham |title= Mean-Value Theorems in the Theory of the Riemann Zeta-Function |journal= Proc. London Math. Soc.|year= 1928|volume= s2-27|issue=1|pages= 273–300| doi=10.1112/plms/s2-27.1.273}}
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| *{{Citation | last1=Ingham | first1=A. E. | title=On the estimation of N(σ,T) | doi=10.1093/qmath/os-11.1.201 | mr=0003649 | year=1940 | journal=The Quarterly Journal of Mathematics. Oxford. Second Series | issn=0033-5606 | volume=11 | issue=1 | pages=291–292}}
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| *{{Citation | last1=Karatsuba | first1=A. A. | last2=Voronin | first2=S. M. | title=The Riemann zeta-function | publisher=Walter de Gruyter & Co. | location=Berlin | series=de Gruyter Expositions in Mathematics | isbn=978-3-11-013170-3 | mr=1183467 | year=1992 | volume=5}}
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| *{{Citation | last1=Keating | first1=Jonathan P. | last2=Snaith | first2=N. C. | title=Random matrix theory and ζ(1/2+it) | doi=10.1007/s002200000261 | mr=1794265 | year=2000 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=214 | issue=1 | pages=57–89}}
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| *{{Citation | last1=Lindelöf | first1=Ernst | title=Quelques remarques sur la croissance de la fonction ζ(s) | year=1908 | journal=Bull. Sci. Math. | volume=32 | pages=341–356}}
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| *{{Citation | last1=Motohashi | first1=Yõichi | title=A relation between the Riemann zeta-function and the hyperbolic Laplacian | url=http://www.numdam.org/item?id=ASNSP_1995_4_22_2_299_0 | mr=1354909 | year=1995 | journal=Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV | issn=0391-173X | volume=22 | issue=2 | pages=299–313}}
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| *{{Citation | last1=Motohashi | first1=Yõichi | title=The Riemann zeta-function and the non-Euclidean Laplacian | mr=1335956 | year=1995 | journal=Sugaku Expositions | issn=0898-9583 | volume=8 | issue=1 | pages=59–87}}
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| *{{Citation | last1=Titchmarsh | first1=Edward Charles | author1-link=Edward Charles Titchmarsh | title=The theory of the Riemann zeta-function | publisher=The Clarendon Press Oxford University Press | edition=2nd | isbn=978-0-19-853369-6 | mr=882550 | year=1986}}
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| *{{eom|id=L/l058960|first=S.M.|last= Voronin}}
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| (The second reference of Voronin's article is false; nothing on the Lindelöf hypothesis is in "Le calcul des résidus et ses applications à la théorie des fonctions")
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| {{L-functions-footer}}
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| {{DEFAULTSORT:Lindelof hypothesis}}
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| [[Category:Conjectures]]
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| [[Category:Zeta and L-functions]]
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| [[Category:Unsolved problems in mathematics]]
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| [[Category:Analytic number theory]]
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a move Godhead directly the speed of light
'Block,ゴルフ用サングラスオークリー!'
Yan Emperor whispered.
immediately to the main hall where he was core to spread out in all directions, a full range of spatial fluctuations of hundreds of light years away completely locked.
'Although I Yan Protoss space coordinates corresponding long concealed to prevent in case, I add some means,オークリー サングラス 調光, even if you invite chaos lord to crack,オークリー サングラス 人気ランキング, it is difficult to crack open short side.' Yan Emperor smiled,オークリー サングラス, a move Godhead directly the speed of light, the shuttle began to cast into the dark universe means it.
,オークリー スポーツサングラス......
Whoosh! Whoosh! Whoosh!
Feng Luo a decision Teleport times, approaching Yan Protoss lair.
'coming to!' Luo Feng increasingly expect.
'ah?' Luo Feng suddenly stopped, looked at the front frown void,ロードバイク サングラス オークリー, the void has some Starshards floating, 'spatial fluctuations even completely blocked? how is it?'
'polytheism ancestral lair,オークリー 激安 サングラス, the lake is unable to teleport colorful aurora,オークリー スポーツサングラス, only some of the island's area Bauhinia island can blink
相关的主题文章：